transcribe vs write down

transcribe

verb
  • To make such a conversion from live or recorded speech to text. 

  • To represent speech by phonetic symbols. 

  • To transfer data from one recording medium to another. 

  • To cause DNA to undergo transcription. 

  • To convert a representation of language, typically speech but also sign language, etc., to a written representation of it. The term now usually implies the conversion of speech to text by a human transcriptionist with the assistance of a computer for word processing and sometimes also for speech recognition, the process of a computer interpreting speech and converting it to text. 

  • To adapt a composition for a voice or instrument other than the original; to notate live or recorded music. 

write down

verb
  • To produce or set (something) down in writing; to record something. 

  • Used other than figuratively or idiomatically: see write, down. 

  • To write (something) in a simple or condescending style. 

  • To make a downward adjustment in the value of an asset. 

  • To condemn in writing; to document the faults, offenses, or wrongdoing of. 

How often have the words transcribe and write down occurred in a corpus of books? (source: Google Ngram Viewer )